Fact 1 (see, e.g., [1]}) 
Given any space \(X\)  and any integer \(n\geqslant -1\) ,
\(\dim X \leqslant n\)  if and only if any finite open cover
\(\lbrace U_1,\dots , U_k\rbrace \)  of \(X\)  has an open shrinking \(\lbrace V_1,\dots , V_k\rbrace \)  of order \(\leqslant n\) .

